Two-variable data: models and scatterplots

This skill point covers data on two variables at once: a scatterplot, a table of paired values, or a fitted model given as an equation. What you are asked to do falls into five jobs, and only five — interpret a slope, interpret an intercept, produce a predicted value, compare a prediction with a measurement (a residual), or decide which shape of model the data actually has. The arithmetic in all five is one multiplication and one addition. The difficulty is entirely in knowing which of the five you were asked for.

College Board's published test specification puts Problem-Solving & Data Analysis at about 15% of the Math section — on the order of five to seven questions. How that splits across the domain's seven skill points is not published, so any claim that you should expect a specific number of scatterplot items is a prep-industry estimate reverse-engineered from released material, not an official figure. Treat it as a planning assumption.

One structural fact makes the whole skill cheaper than it looks: the model is always handed to you. Meridian has not seen a released Digital SAT item that asks a student to compute a line of best fit by hand, and there is no formula for one on the reference sheet. You get an equation, a described line, or a graph with the line already drawn, and the work is reading it. That is why this skill rewards care rather than technique.

What is genuinely new here, compared with the linear-equation and linear-function skills next door, is the gap between model and measurement. Everywhere else on the Math section the equation is the truth. Here the equation is a summary of something messier, the dots do not sit on the line, and the exam has a question type built specifically on that distance.

If you can look at a scatterplot with a fitted line and immediately say what one dot means, what the line means, and how far apart they are, skip ahead. If any of that is fuzzy, this is the highest-value block on the page, because everything later is a sentence about these objects.

A scatterplot shows one thing measured twice. Each dot is a single case — one city, one student, one plant, one year — and its horizontal position is one measurement of that case while its vertical position is the other. Twenty dots means twenty cases, not twenty numbers.

Read the axes before the dots, every time. The axis labels carry the whole meaning, and two riders on them cause more lost points than anything else in this skill: a shifted origin ("years since 1995", so 1998 is x = 3, not x = 1998) and a scaling rider ("population, in thousands", so a height of 46.5 means 46,500 people).

Association is the pattern in the cloud. If the dots drift upward as you move right, the association is positive; downward, negative; if they hug a pattern tightly, it is strong, and if they scatter loosely around it, weak. Association is not causation — two quantities can move together because one drives the other, because something else drives both, or by accident, and a scatterplot cannot tell those apart.

The line of best fit is a single straight line drawn through the cloud to summarize that pattern. Three things about it are worth fixing now. It is a summary, not a boundary — dots are expected on both sides of it. It is not required to pass through any dot at all, and usually passes through none of them. And it is a claim about the trend, so it can be a perfectly good line through data that no straight line describes well.

That gives the distinction the entire skill runs on. The dots are ACTUAL values: what was measured. The line gives PREDICTED values: what the model says to expect at that x. The exam signals which one it wants with the words "predicted", "the model", "the line of best fit" on one side, and "measured", "recorded", "the data point", "actual" on the other. The exam writes the model as y = 812 − 6.4x, which quietly hides that this y is a prediction; when you copy a model onto your scratch paper, writing it as ŷ = 812 − 6.4x costs a second and keeps the two apart for the rest of the question.

Getting a predicted value out of the model: substitute the x you were given. From a graph rather than an equation, find the x on the horizontal axis, go up to the line — not to a nearby dot — and read across to the vertical axis.

The slope of the line of best fit is a rate in the same sense it is everywhere else: y-units per one x-unit, carrying the sign of the direction. Compute it from two points that lie ON THE LINE. Two data points give you the slope of the segment joining two dots, which is a different line and almost never the same number.

The y-intercept of the line of best fit is the predicted y when x = 0, exactly as in any linear equation. Whether that number means anything in the real situation depends on whether x = 0 is inside the range of the data — a question taken up in the 800-level block, because it is where strong students lose this item.

A residual is the gap between one dot and the line, straight up or down: residual = actual − predicted. Positive means the dot is above the line and the model underestimated that case; negative means the dot is below the line and the model overestimated it. It is measured in the y-variable's units, and it belongs to one data point, not to the model as a whole.

A curve of best fit is the same idea with a curved model instead of a straight one. On the Digital SAT the curved model is almost always exponential: y = a·b^x, where a is the predicted value at x = 0 and b is the multiplier per one x-unit. b greater than 1 is growth, b between 0 and 1 is decay, and the percent change per step is (b − 1) expressed as a percentage — 1.2 is a 20% increase, 0.85 is a 15% decrease.

Step 1 — name x and y with units, in writing, including the riders. "x: years after 2005. y: subscribing households, in thousands." Ten seconds, and it is what makes the difference between an answer of 24 and an answer of 24,000 at the end.

Step 2 — decide which of the five jobs the question is asking for, before computing anything. Is it a RATE (the slope, per one x-unit)? A LEVEL (a predicted value of y at a stated x)? A CHANGE OVER AN INTERVAL (the slope times the interval's length)? An INPUT (the x that produces a stated y, so solve rather than substitute)? Or a RESIDUAL (a comparison between one measurement and the model)? The four you were not asked for are, reliably, three of the four answer choices.

Step 3 — for a slope interpretation, say it as a full sentence with units and a direction before looking at the options: "for each additional [one x-unit], the predicted [y-quantity] increases/decreases by [number] [y-units]." A choice that omits "predicted", or reports the number without the per-one-unit, or drops the direction, has failed a part of that sentence.

Step 4 — for an intercept interpretation, say: "when x is 0, the model predicts y = b." Then translate x = 0 back into the situation — the year 2005, the day of arrival, zero minutes of use — because the answer choices will be phrased in the situation's language, not in x.

Step 5 — for a residual, subtract in the fixed order, actual minus predicted, then check it with the identity predicted + residual = actual. Then translate the sign into English before you answer, because the question often asks for "how much the model overestimates" rather than for a signed number.

Step 6 — for a model-shape decision, run two rows of arithmetic across the table. Constant DIFFERENCES between consecutive y-values (at evenly spaced x) means linear. Constant RATIOS means exponential. If the x-values are not evenly spaced, compare rates over each interval instead of raw differences. And if the description is verbal rather than tabular, the tell is the same one in words: a constant AMOUNT per step is linear, a constant PERCENTAGE per step is exponential.

By this point the method is not what separates 1500 from 1600 on this skill. Six things are: the predicted/actual distinction hidden inside ordinary wording, residual plots, exponential models whose time step is not one unit, intercepts that are mathematically fine and physically meaningless, the interval-versus-level family of questions, and a short list of execution errors that survive knowing all of the above.

The wording, first, because it is worth the most. An item does not have to use the word "residual" to be a residual item. "By how much does the model overestimate the value for that city", "how much greater was the recorded amount than the amount the line predicts", "the data point lies how far above the line of best fit" are all the same computation in different clothes. The reverse is also written: an item that mentions a specific measured value but asks only for the model's prediction is testing whether you will subtract when nothing asked you to. Underline which of the two objects — the dot or the line — the final sentence names.

Residual plots. A residual plot puts x on the horizontal axis and the residual on the vertical, with a horizontal line at zero. Read the shape, not the size. Randomly scattered above and below zero with no pattern means the straight-line model captured the relationship. A run of positives, then negatives, then positives — a U or an arch — means the true relationship is curved and a linear model is systematically wrong at the ends and in the middle, no matter how good the correlation looks on the original plot. A fan that widens as x grows means the spread of the data changes with x, so predictions get less trustworthy toward the wide end. And the sum of the residuals is zero for any least-squares line with an intercept, which makes a zero sum an arithmetic identity rather than evidence of anything — a choice offering it as proof of good fit is offering you a true statement that is not relevant.

Exponential models with a non-unit time step. y = a·b^x claims a multiplier of b per ONE x-unit, so anything described per five hours, per decade, or per half-life must be written with the step inside the exponent. A culture that doubles every 5 hours, with h in hours, is N(h) = N₀·2^(h/5) — not 2^(5h), which doubles five times an hour and is off by a factor of millions within a day. A substance with a 12-year half-life is A(t) = A₀·(1/2)^(t/12), equivalently about A₀·(0.944)^t per year, because 0.5^(1/12) ≈ 0.944 — roughly a 5.6% annual decline. Both forms are correct and the exam will print whichever one you did not write, so recognizing the equivalence is faster than re-deriving it.

Meaningless intercepts. A model of heating cost against average January temperature over a range of −4 to 21 °C has x = 0 sitting comfortably inside the data, so its intercept is a genuine prediction. A model of a child's height against age fitted over ages 2 to 10 has an intercept of about 84 cm at age 0, which is not any newborn's height — it is the number a straight line happens to hit when extended somewhere it was never tested. Both intercepts are correct arithmetic. Only one is a fact about the world, and the exam writes items in which the correct choice is an interpretation of the model rather than a claim about reality. Read the choice's verb: "the model predicts" is safe, "the value was" is a claim the data may not carry.

Interval versus level, in the two directions the exam uses. Forward: given the model, the change over any k x-units is slope × k, and the level at a stated x is the substitution — different numbers, both offered. Backward: "the model predicts a fall of 900 over the fifteen-year period, so what is the annual rate" is a division, and "in what year does the model first predict a value below 400" is a solve followed by a translation from t back into a calendar year. That final translation is where the point goes: t = 13 with x defined as years after 1998 is the year 2011, and 13 will be sitting in the choices.

One more that catches strong students: two models on one plot, or one model and one summary statistic. If a plot carries a line of best fit and a horizontal line at the mean, or two fitted lines for two groups, every question names one of them and the other's answer is a choice. The same applies to items pairing this skill with margin of error or a sample statistic — the model answers a different question from the interval, and reading which is being asked takes less time than recovering from having answered the other.

Execution errors, which is where the last few points actually live. Rounding a slope before the final multiplication, when 28/8 is exactly 3.5 and 0.33 is not exactly a third. Reporting the predicted value when the residual was asked for, or the residual when the predicted value was asked for. Answering in thousands because the axis was in thousands. Off-by-one on an interval of years. Substituting a year where the model wants years-since. Every one of these produces a number that is on the answer sheet, which is why re-reading the final sentence before selecting is worth more here than anywhere else in the domain — Bluebook gives you Desmos on every Math question and it will graph the model happily, but it cannot tell you which quantity the sentence asked for.